If the register icon is missing, then that means that mIRC is being recognized as being registered on that computer.
You said that you're re-downloading mIRC. Does this mean that you've downloaded, installed and registered mIRC on that system before, or are you downloading it for the first time?
If you go to Help - About, it will show in the pop-up dialog box the name of the person that mIRC is recognizing as being registered to. If it's not recognizing a registration, then it will show as Unregistered.
Please note that registration of mIRC has nothing to do with the nick that you use for chatting on the IRC networks, and any registration of that nick.